Classical logic is a system concerned with certain objects that can attain either of two values (usually interpreted as

*propositions*that may be either

*true*or

*false,*commonly denoted 1 or 0 for short), and ways to connect them. Though its origins can be traced back in time to antiquity, and to the Stoic philosopher Chrysippus in particular, its modern form was essentially introduced by the English mathematician and philosopher George Boole (and is thus also known under the name

*Boolean algebra*) in his 1854 book

*An Investigation of the Laws of Thought*, and intended by him to represent a formalization of how humans carry out mental operations. In order to do so, Boole introduced certain

*connectives*and

*operations*, intended to capture the ways a human mind connects and operates on propositions in the process of reasoning.

An elementary operation is that of

*negation*. As the name implies, it turns a proposition into its negative, i.e. from 'it is raining today' to 'it is

*not*raining today'. If we write 'it is raining today' for short as

*p*, 'it is not raining today' gets represented as ¬

*p*, '¬' thus being the symbol of negation.

Two propositions,

*p*and

*q*, can be connected to form a third, composite proposition

*r*in various ways. The most elementary and intuitive connectives are the

*logical and*, denoted by ˄, and the

*logical or,*denoted ˅.

These are intended to capture the intuitive notions of 'and' and 'or': a composite proposition

*r*, formed by the 'and' (the

*conjunction*) of two propositions

*p*and q

*,*i.e.

*r = p*˄

*q*, is true if both of its constituent propositions are true -- i.e. if

*p*is true

*and*

*q*is true. Similarly, a composite proposition

*s*, formed by the 'or' (the

*disjunction*) of two propositions

*p*and

*q*, i.e.

*s = p ˅ q*, is true if at least one of its constituent propositions is true, i.e. if

*p*is true

*or q*is true. So 'it is raining and I am getting wet' is true if it is both true that it is raining and that you are getting wet, while 'I am wearing a brown shirt or I am wearing black pants' is true if I am wearing either a brown shirt or black pants -- but also, if I am wearing both! This is a subtle distinction to the way we usually use the word 'or': typically, we understand 'or' to be used in the so-called

*exclusive*sense, where we distinguish between two alternatives, either of which may be true, but not both; however, the logical 'or' is used in the

*inclusive*sense, where a composite proposition is true also if both of its constituent propositions are true.